Normal Distribution, Central Limit Theorem, Gamma Distribution

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Lecture 9

• Normal Distribution, Central Limit Theorem, Gamma
  Distribution

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• The Reproductive Property of Normal Distribution

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• Examples
• Three steel pipes are selected randomly from
  N(12.00, 0.0016) population and assembled
  end-to-end. Compute the probability that the
  total length of the assembly exceeds 36.12?.
• n3, each ci 1, and each Xi N(12.00, 0.0016)

Suppose 4 steel pipes are selected at random from
the N(12, 0.0016) population and placed
end-to-end. (a) Compute the pr that the length of
the assembly does not exceed 47.90". (b) Compute
the pr that the average length of the 4 pipes
exceeds 12.04.
(a) n4, each ci 1, and each X1 (12.00,
0.0016)
(b) n4, each ci 1/4, and each X1 (12.00,
0.0016)
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• The Central Limit Theorem

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• Central limit theorem

Refer to the The Moments of a Random
Variable.doc handout for the values of ?3 and ?4.
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• The Normal Approximation to the Binomial

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• Example
• Consider a binomial distribution with n50 trials
  and p0.30. (So that ?np15). Note that we would
  like to have npgt15, but n50gt25(?3)2 is easily
  satisfied. Then the Range space of the discrete
  rv XD0,1,2,,50. The exact probability of
  attaining exactly 12 successes in 50 trials is
  given by b(12 50, 0.30) P(XD12) 0.08383.
• To apply the normal distribution approximation
  (with ?15 and ?2npq10.50), we first need to
  select an interval on a continuous scale to
  represent the discrete XD12. Clearly, this
  interval needs to be (11.5,12.5)C . Thus,
  P(XD12) P(11.5ltXClt12.5)

8
0.08383
Gray Area 0.080157
XD12
XC11.5
XC12.5
9
Green Area 0.77980
0.08383
Gray Area 0.080157
XD12
XC11.5
XC12.5
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• The Gamma Distribution and its Relatives
• The Gamma Function

• Properties of the Gamma function
• For any ?gt1, ?(?) (??1) ?(??1)
• For any positive integer, n, ?(n) (n?1)!
• A continuous rv X has a Gamma Distribution if the
  pdf of X is

Where ?gt0 and ?gt0. The standard Gamma
distribution has ?1
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• Why do we need gamma distribution?
• Any normal distribution is bell-shaped and
  symmetric. There are many practical situations
  that do not fit to symmetrical distribution.
• The Gamma family pdfs can yield a wide variety of
  skewed distributions.
• ? is called the scale parameter because values
  other than 1 either stretch or compress the pdf
  in the x-direction.

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• The mean and the variance of a gamma distribution

• The cdf of the standard gamma rv X

is called the incomplete gamma function. There
are extensive tabulations available for F(x?).
We will refer to page 742 on Devore, for ?
1,2,,10 and x 1,2,..,15.
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• Example
• The reaction time X of a randomly selected
  individual to a certain stimulus has a standard
  gamma distribution with ?2.
• Compute the probability that the reaction time
  is between 3 and 5 time units.
• P(3?X?5) F(52) ? F(32) 0.960 ? 0.801
  0.159
• Compute the probability that the reaction time
  is longer than 4 time units.
• P(Xgt4) 1 ? F(42) 1 ? 0.908 0.092

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• Let X have a gamma distribution with parameters ?
  and ?. For any xgt0, the cdf of X is given by

where F(??) is the incomplete gamma function.
Example X the survival time in weeks. ?8,
?15. Expected survival time E(X)(8)(15)120
weeks. V(X) (8)(15)2 1800, ?x
42.43 P(60?X?120) P(X?120) - P(X?60)
F(120/158) F(60/158) F(88) F(48)
0.547 0.051 0.496 P(X?30) 1 P(Xlt30) 1
P(X?30) 1 F(30/158) 0.999